The long-term objective of the on-going research is to understand completely the dynamics of any populations without the usual constraints of a stable population. It was usually assumed that the dynamics of one-sex, closed populations with unchanging age- specific vital rates are completely specified. However, it became obvious in the course of research on the dynamics of populations with changing vital rates that there remain fundamental problems to be resolved for populations with fixed vital rates. More specifically, while the stable limit is known when vital rates are constant, the understanding of the determinants of the rate at which a population reaches stability was found to be far from being complete. The proposed research is to fill this important and fundamental gap first, then to generalize the findings to populations with the constraints removed. The first specific aim is to characterize various (at least eight) measures of the distance from stability and understand relationships among the measures. Only when the measures of the distance are fully specified, can the search for the determinants of the speed of convergence be addressed. The next and major aim of the proposed research is to search for the complete determinants of the speed of convergence to stability. These aims will be achieved by both empirical and analytical methods. The empirical study will utilize extensively the data on human populations compiled by Keyfitz and Flieger. For the analytical study, the Kim representation for the powers of the population projection matrix developed over the recent years will play an important role. When the firts two aims are achieved, the parallel research will be carried out for populations with arbitrarily changing vital rates. The next extension will be to multistate populations relaxing the constraint of closed populations. The final extension will be to include both aspects of extensions by considering multistate populations with arbitrarily changing vital rates.